Bring milk just to a boil in a 2-quart heavy saucepan. Whisk together eggs and sugar in a large bowl, then add hot milk in a slow stream, whisking. Pour mixture into saucepan and cook over moderately low heat, stirring constantly with a wooden spoon, until mixture registers 170°F on thermometer, 6 to 7 minutes. Pour custard through a fine-mesh sieve into cleaned large bowl and stir in cream, bourbon, brandy, and vanilla. Cool completely, uncovered, then chill, covered, until cold, at least 3 hours and up to 24.

Note: Flavour of eggnog improves when it is made a day ahead to allow alcohol to mellow.